2. Why is funding local police the federal government’s purview.
Because they want a national police force like the French Gendarmerie Nationale - which specifically takes care of the areas outside Paris.
https://en.wikipedia.org/wiki/National_Gendarmerie
The easier to take them over.
To your question...why is funding the local police a federal government purview?
Same question...why don’t they fund dog-catchers, Little League Baseball across the nation, honky-tonks, and all 3rd-grade school teachers?
Once you open up this mess, we all contribute into a pot to help poor blue-states, which failed to keep businesses open and keep revenue taxes at a level to cover simple things like local police.